Adsorption Isotherm Fitting and Breakthrough Curve Modeling for Fixed-Bed Columns (Langmuir, Freundlich, Thomas, Yoon–Nelson)
The purpose of this article is to provide a practical, technically rigorous workflow to fit adsorption isotherms (Langmuir and Freundlich) and to model fixed-bed breakthrough curves with both simple analytical models and a physics-based axial-dispersion plus LDF framework, so you can extract defensible parameters for design and scale-up.
1. Why isotherm fitting and breakthrough modeling must be treated as one problem.
In adsorption process development, equilibrium capacity and affinity determine whether the adsorbent can carry the load, while mass transfer and hydrodynamics determine whether the bed can deliver that capacity at the required flow rate and service time.
Isotherm fitting converts batch equilibrium data into a function q*(C,T) that predicts the equilibrium loading at the particle surface.
Breakthrough curve modeling converts column time-series data into transport and kinetic parameters that explain how fast the bed approaches q*(C,T) along the axial direction.
When you fit an isotherm and then independently fit a breakthrough model that implicitly assumes a different equilibrium relationship, you can obtain parameters that look numerically good but are not physically consistent.
A robust workflow therefore uses isotherm fitting to define q*(C) and then uses that same q*(C) inside the column model whenever possible.
2. Data requirements and preparation.
2.1 Batch equilibrium data checklist for adsorption isotherm fitting.
Prepare equilibrium pairs (Ce, qe) at a fixed temperature, where Ce is the equilibrium fluid concentration and qe is the adsorbed amount per mass of adsorbent.
Use consistent units and keep them explicitly in your dataset, because fitted constants carry units that must remain consistent in downstream modeling.
Typical choices are C in mg/L or mol/m3, and q in mg/g or mol/kg.
For liquid-phase systems, ensure mass balance closure, account for sampling losses, and correct for any blank adsorption to container walls.
For gas-phase systems, record pressure, temperature, and partial pressure explicitly, and convert to consistent concentration or pressure basis as needed.
2.2 Fixed-bed column data checklist for breakthrough curve modeling.
At minimum, you need influent concentration C0, effluent concentration C(t), volumetric flow rate Q, bed dimensions (diameter and packed height Z), adsorbent mass m, and operating temperature.
Record the time origin carefully, because dead volume and sampling lines can shift the apparent curve.
Report C/C0 versus time t, and also compute bed volumes processed BV = Q t / Vbed to compare runs across different bed sizes.
| Dataset type | Minimum measured variables | Recommended additional variables | Primary output |
|---|---|---|---|
| Batch equilibrium | Ce, qe, T | pH, ionic strength, competing solutes, replicate errors | Isotherm parameters and confidence intervals |
| Fixed-bed breakthrough | C(t), C0, Q, Z, m, T | Particle size, void fraction, pressure drop, viscosity, density | Breakthrough model parameters and design metrics |
3. Langmuir adsorption isotherm fitting.
3.1 Langmuir model definition and interpretation.
The single-site Langmuir isotherm is widely used when adsorption approaches a finite monolayer capacity.
The standard form is:
q = (qmax * b * Ce) / (1 + b * Ce)Here q is the equilibrium loading, qmax is the saturation capacity, b is the affinity constant, and Ce is the equilibrium fluid concentration.
At low Ce, the model approaches q ≈ qmax b Ce, so the initial slope is qmax b.
At high Ce, the model approaches q → qmax, which provides a physically meaningful ceiling for capacity.
3.2 Linearizations and why they can mislead.
Common linear forms include Ce/q versus Ce, and 1/q versus 1/Ce.
For example:
Ce/q = (1/(qmax*b)) + (Ce/qmax)Linearization can be useful for quick initial guesses, but it changes the error structure, over-weights low concentration points, and can bias qmax and b when measurement errors are not uniform.
For design-grade parameters, non-linear regression on the original equation is preferred.
3.3 Non-linear regression workflow for Langmuir fitting.
Step 1 is to choose an objective function, typically the sum of squared residuals in q.
Step 2 is to select a weighting scheme if the variance grows with q or Ce, such as weighting by 1/q2 or using experimentally estimated standard deviations.
Step 3 is to fit qmax and b with constraints that keep both positive.
Step 4 is to compute parameter confidence intervals using the covariance estimate from the Jacobian of the residuals.
Step 5 is to validate the fit by inspecting residuals versus Ce rather than relying only on R2.
3.4 Separation factor for quick feasibility checks.
A commonly used dimensionless indicator for Langmuir favorability is:
RL = 1 / (1 + b * C0)Here C0 is the influent or initial concentration relevant to the process.
Values 0 < RL < 1 indicate favorable adsorption on that concentration range, while RL → 0 indicates very strong affinity.
4. Freundlich adsorption isotherm fitting.
4.1 Freundlich model definition and interpretation.
The Freundlich isotherm is an empirical power-law model often used for heterogeneous surfaces and moderate concentration ranges where saturation is not clearly observed.
The standard form is:
q = KF * Ce^(1/n)Here KF is the Freundlich capacity factor and 1/n represents adsorption intensity or heterogeneity.
If 1/n is closer to 1, the isotherm is closer to linear partitioning, while smaller 1/n indicates stronger nonlinearity.
4.2 Log-linear form and fitting cautions.
A common linear form is:
log(q) = log(KF) + (1/n) * log(Ce)This form is convenient, but it assumes multiplicative errors and can bias parameters when q includes zeros, negative values after blank correction, or when the error variance is not proportional to q.
As with Langmuir, direct non-linear regression on the original equation is preferred for design-grade use.
4.3 When Freundlich is not appropriate.
If your data show a clear plateau at high Ce, Freundlich will not capture saturation and will overpredict capacity at higher concentrations.
In that case, Langmuir-type models or other saturating isotherms are a better physical choice.
| Model | Equation | Strength | Common limitation |
|---|---|---|---|
| Langmuir | q = (qmax b Ce)/(1 + b Ce) | Finite saturation capacity, interpretable qmax | May not capture strong heterogeneity without extensions |
| Freundlich | q = KF Ce^(1/n) | Flexible for heterogeneous surfaces in mid-range Ce | No saturation, unreliable extrapolation to high Ce |
5. How to report isotherm fitting results so they are usable in design.
Report fitted parameters with units, confidence intervals, temperature, and the exact fitting method used.
Include the dataset range, because extrapolation outside that range is the largest source of silent design errors.
Provide at least one goodness-of-fit metric that is meaningful for non-linear models, such as RMSE, and include a residual plot summary in words.
State whether the fit used unweighted least squares or a weighting scheme.
6. Breakthrough curve fundamentals for fixed-bed adsorption.
6.1 Breakthrough curve definitions.
The breakthrough curve is typically plotted as C/C0 versus time t or bed volumes BV.
The breakthrough time tB is often defined at C/C0 = 0.05 or 0.10, and the exhaustion time tE is often defined at C/C0 = 0.90 or 0.95.
The mass transfer zone concept describes the moving region of partial saturation that travels down the bed.
A narrow mass transfer zone produces a steep curve and high bed utilization, while a wide zone produces an early breakthrough and long tail.
6.2 Capacity from the breakthrough curve by mass balance.
For a single solute with negligible accumulation in the fluid voids, the total captured mass up to time t is obtained by integrating the deficit between influent and effluent.
m_adsorbed(t) = Q * ∫[0→t] (C0 - C(t)) dtThe corresponding average loading on the adsorbent is:
q_avg(t) = m_adsorbed(t) / mAt exhaustion, q_avg should be consistent with the equilibrium capacity predicted from your isotherm at the local conditions, acknowledging that real beds seldom reach perfect equilibrium everywhere.
7. Analytical breakthrough models used for rapid parameter extraction.
7.1 Thomas model.
The Thomas model is commonly used as a quick logistic-type representation for C/C0 versus time and can yield an apparent capacity parameter and a kinetic constant.
A widely used linearized form is:
ln((C0/C) - 1) = (kTh * q0 * m / Q) - (kTh * C0 * t)Here kTh is the Thomas rate constant, q0 is the apparent adsorption capacity in the column, m is adsorbent mass, Q is volumetric flow rate, and t is time.
Plotting ln((C0/C) - 1) versus t can provide initial estimates, but non-linear fitting on the original logistic expression is preferred for accurate uncertainty estimates.
7.2 Yoon–Nelson model.
The Yoon–Nelson model is a two-parameter approach that focuses on the time τ when 50% breakthrough occurs.
A common form is:
ln(C/(C0 - C)) = kYN * (t - τ)Here kYN is the rate constant and τ is the time corresponding to C/C0 = 0.5.
This model is useful when you want a compact description of curve steepness and mid-point timing without explicitly estimating qmax.
7.3 Bohart–Adams and BDST for bed-height scaling.
The Bohart–Adams model is frequently used to relate breakthrough to bed depth at early times, and BDST uses that relationship to produce a design line for required bed height at a target service time.
A commonly used BDST expression is:
t = (N0 * Z) / (C0 * v) - (1 / (kBA * C0)) * ln((C0/Cb) - 1)Here t is service time to reach a breakthrough concentration Cb, Z is bed depth, v is superficial velocity, N0 is a volumetric capacity term, and kBA is a kinetic constant.
BDST is valuable for rapid scale-up across bed heights when operating conditions remain similar and when the early part of the curve is the main design constraint.
7.4 How to choose among simple models.
Use the Thomas model when you want an apparent capacity and a single steepness parameter for symmetric S-shaped curves.
Use Yoon–Nelson when you primarily care about the time to 50% breakthrough and want a minimal parameter set.
Use BDST when you have multiple bed heights and need a service-time versus bed-depth design correlation.
If your curve is strongly asymmetric, shows a long tail, or changes shape with flow rate, consider a physics-based model with dispersion and mass transfer rather than relying only on simple analytical equations.
| Model | Typical fitted parameters | Best use case | Main assumption to watch |
|---|---|---|---|
| Thomas | kTh, q0 | Quick capacity and curve steepness for near-symmetric curves | Often assumes negligible axial dispersion and a logistic shape |
| Yoon–Nelson | kYN, τ | Compact description centered on 50% breakthrough | Empirical shape, may not capture strong asymmetry |
| BDST | N0, kBA | Bed-height scaling at a chosen breakthrough criterion | Validity mainly for the initial breakthrough region |
8. Physics-based breakthrough modeling with axial dispersion and LDF kinetics.
8.1 Core equations for a 1D packed bed.
A common isothermal one-dimensional description couples a fluid-phase balance with an adsorbed-phase rate expression.
A representative form is:
ε * ∂C/∂t + u * ∂C/∂z = Dax * ∂²C/∂z² - (1 - ε) * ρb * ∂q/∂tHere ε is bed void fraction, u is interstitial or superficial velocity (use a consistent definition), z is axial position, Dax is axial dispersion coefficient, ρb is bed bulk density, and q is solid loading.
The linear driving force approximation then relates the rate of uptake to the driving force from the equilibrium loading q*:
∂q/∂t = kLDF * (q* - q)The key point is that q* is computed from your fitted isotherm, such as Langmuir or Freundlich.
This allows you to combine equilibrium (q*) and kinetics (kLDF) into a consistent model.
8.2 Boundary conditions and practical interpretation.
At the inlet, a Danckwerts-type boundary condition is often used to include dispersion consistently:
Dax * (∂C/∂z)|inlet = u * (C - C0)At the outlet, a zero-gradient condition is often used:
(∂C/∂z)|outlet = 0In practice, Dax controls front spreading, while kLDF controls how quickly the adsorbent loading follows the equilibrium loading q* behind the front.
8.3 Parameter estimation strategy that avoids non-identifiability.
If you fit both Dax and kLDF from a single breakthrough curve, they can compensate for each other, because both can broaden the front.
To reduce ambiguity, estimate Dax from hydrodynamic correlations or from a non-adsorbing tracer test, and then fit kLDF from adsorption breakthrough data.
Alternatively, fit multiple curves at different flow rates, because dispersion and mass transfer respond differently to changes in velocity.
Use the same isotherm parameters across all curves at the same temperature, and only vary transport parameters if there is a physical reason to do so.
9. A step-by-step workflow you can apply to real adsorption datasets.
9.1 Step 1: Fit Langmuir and Freundlich to equilibrium data.
Fit both models with non-linear regression on q versus Ce using the same weighting philosophy.
Record parameter uncertainties and confirm that the fitted curve does not violate obvious physical behavior in your concentration range.
Decide which isotherm to use for column modeling based on residual structure and extrapolation needs, not only on a single goodness-of-fit number.
9.2 Step 2: Extract design metrics directly from breakthrough curves.
Compute tB and tE at chosen C/C0 criteria, compute treated bed volumes at breakthrough, and compute captured mass by integration.
These metrics are model-free and should be consistent across any model you later fit.
9.3 Step 3: Fit a simple analytical model for fast comparisons.
Fit Thomas and Yoon–Nelson to obtain comparable steepness and timing parameters across different operating conditions.
Use BDST if you have multiple bed heights and want a fast bed-depth correlation for a chosen breakthrough criterion.
9.4 Step 4: Fit a physics-based model for scale-up and scenario prediction.
Implement an axial dispersion plus LDF model using your chosen isotherm for q*(C).
Fit kLDF and, if necessary, Dax using multiple runs to stabilize parameter estimates.
Validate the model by predicting a condition that was not used in fitting, such as a different bed height or flow rate.
9.5 Step 5: Report parameters in a design-ready format.
Provide qmax and b or KF and 1/n with units and temperature.
Provide kTh and q0 or kYN and τ if using simple models.
Provide kLDF and Dax with the velocity definition used, and state whether Dax was fixed or fitted.
10. Illustrative example structure for documenting results.
The table below shows an example layout you can use to report fitted results without mixing unit systems.
| Category | Parameter | Symbol | Unit | How obtained | Used for |
|---|---|---|---|---|---|
| Langmuir isotherm | Maximum capacity | qmax | mg/g (or mol/kg) | Non-linear regression on q vs Ce | Equilibrium loading q*(C) |
| Langmuir isotherm | Affinity constant | b | L/mg (or m³/mol) | Non-linear regression on q vs Ce | Equilibrium loading q*(C) |
| Freundlich isotherm | Capacity factor | KF | (mg/g)(L/mg)^(1/n) | Non-linear regression on q vs Ce | Alternative q*(C) form |
| Freundlich isotherm | Intensity | 1/n | dimensionless | Non-linear regression on q vs Ce | Curvature and heterogeneity |
| Breakthrough (simple) | Thomas rate constant | kTh | mL/(mg·min) or L/(mg·min) | Fit C/C0 vs t | Quick curve comparison |
| Breakthrough (physics) | LDF mass-transfer coefficient | kLDF | 1/s | Fit axial model using q*(C) | Scale-up predictions |
| Breakthrough (physics) | Axial dispersion coefficient | Dax | m²/s | Tracer test or multi-curve fit | Front spreading prediction |
11. Common failure modes and how to fix them.
11.1 Overfitting the breakthrough curve with too many free parameters.
If you fit qmax, b, kLDF, and Dax from a single curve, you can often obtain many parameter sets with similar visual fits.
Fix the isotherm from batch data, constrain Dax using hydrodynamics or tracer tests, and fit only the minimal set needed to match curves across multiple runs.
11.2 Using inconsistent concentration bases between isotherm and column models.
If your isotherm was fitted with Ce in mg/L but the column solver uses mol/m3, the model will silently fail unless all parameters are converted consistently.
Always carry units with each parameter in your documentation and keep one canonical internal unit system for computation.
11.3 Ignoring temperature and competitive adsorption effects.
Isotherm parameters are temperature-dependent and can shift with ionic strength, pH, and competing solutes.
If the feed matrix changes, treat q*(C) as a different function and re-fit or build a multicomponent model rather than reusing single-solute parameters without verification.
11.4 Misinterpreting good curve fit as proof of mechanism.
Simple analytical models can fit many curve shapes because they are flexible, not because they are mechanistic.
Use them for comparison and rapid screening, and rely on physics-based models for extrapolation across scale, bed geometry, and operating conditions.
FAQ
Should I use linearized Langmuir and Freundlich plots or non-linear regression.
For quick visualization and initial guesses, linear plots are acceptable.
For design-grade parameters and meaningful uncertainty estimates, non-linear regression on the original isotherm equations is preferred because linearization changes the error structure and can bias parameters.
Why does my breakthrough curve look asymmetric with a long tail.
A long tail can be caused by axial dispersion, slow intraparticle diffusion, film resistance, non-uniform flow, particle size distribution, or non-ideal equilibrium such as multiple adsorption sites and competitive effects.
If asymmetry is strong, consider an axial dispersion plus LDF model with a properly fitted isotherm, and verify hydrodynamics with a tracer test.
How do I select the breakthrough criterion for design.
Select C/C0 based on the downstream quality requirement, such as a regulatory limit, product specification, or catalyst poison threshold.
Document the chosen criterion explicitly, because BDST and service-time calculations depend directly on this definition.
Can I scale up from lab column to pilot column using only the Thomas model.
Thomas-type models can provide trend guidance but often fail when bed diameter, flow distribution, particle size, and hydrodynamics change.
For scale-up, a physics-based model that includes dispersion and mass transfer, anchored by isotherm data, is more reliable for extrapolation.
What is the minimum dataset for a defensible adsorption model.
A practical minimum is a batch equilibrium isotherm at the operating temperature and at least two breakthrough curves at different flow rates or bed heights under the same feed composition.
This combination helps separate equilibrium from kinetics and reduces parameter non-identifiability.